Decisions & Learning in
Computational
Behavioral
Game Theory
decisions in
game theory
Ui(X)=xi
Players only care about their
own payoffs. This is elegant, but
wrong. A host of experiments
show that people don’t make
economic decisions this way.
ultimatum bargaining
Despite some limitations in explaining economic behavior, “classical” game
theory as developed by von Neumann, Nash, and others has proved an
extremely powerful tool in economics. Certain obvious applications of game
theory are impossible, however. Imagine a matrix game representing
international trade. All the countries of the world are players, and each
player’s payoffs depend on the
moves of all the players. To
extremely simplify the possible actions, let each country simply choose one
other country to focus on in improving exports. The U.S. Department of
State lists 192 countries. Even if 50 of these can be neglected, the matrix
for this game would have 142142 entries — substantially more entries than
there are particles in the universe. We cannot even write down the
specification of this game, much less begin to compute its Nash equilibrium.
To deal with such problems, researchers in computational game theory have
sought approximate solutions and have rewritten the problem to take into
account the structure of interactions among players.
Player 1 chooses how to split $10. Player 2 then
accepts this split (both players take their share) or
rejects it (neither player gets anything). That’s all.
Player 2 should accept any split greater than $0,
even accepting only 25¢ or 1¢. So player 1 should
offer almost nothing, so as to get the most money
from the game. But hundreds of experiments in
different cultures, with different stakes, show that
people don’t behave this way. Almost always, low
offers are rejected. Here, game theory doesn’t
predict what players do; it also doesn’t offer good
advice about what to do when playing with a person.
game theory + computation
decisions in behavioral game theory
β
α
Ui(X)=xi - n-1
Σmax(xk-xi,0) - n-1
Σmax(xi-xk,0)
k=
/i
k=
/i
Players care about their own payoffs, they dislike others having more than them with
coefficient α, and they dislike having more than others with coefficient β. This
particular model (Fehr and Schmidt, 1999) is one of several behavioral theories.
I am working now with Michael Kearns and other researchers to uncover
and address important questions at the intersection of computational
game theory, behavioral economics, and machine learning. A new
behavioral game theory is now in development, seeking to explain how
people make economic decisions (as actually observed in experiments)
and how they learn to reach the observed equilibria.
Many important questions remain regarding the computational aspects
of the new, behavioral models: As the number of players and actions
becomes large, is the computational complexity of problem similar to that
of the classical game theory problem? Are the suggested learning
algorithms tractable for large numbers of players or actions? I will also
consider appropriate ways that graphical models can be used to encode
local structure in behavioral game theory.
Camerer, Colin, Teck-Hua Ho, and Juin Kuan Chong. “Behavioral Game Theory: Thinking,
Learning, and Teaching.” Available online. 14 November 2001.
Fehr, Ernst and Klaus Schmidt. “A Theory of Fairness, Competition, and Cooperation.”
Quarterly Journal of Economics 114. 817-868. 1999.
learning in
behavioral game theory
j
j
j
φN(t-1)Ai(t-1)+[δ+(1-δ)I(si,si(t))]πi(si,si(t))
j
Ai(t)=
N(t-1)φ(1-κ)+1
N(t)=N(t-1)φ(1-κ)+1
One model of how economic behavior changes over time is
FEWA, functional experience-weighted attraction learning
(Camerer, Ho, and Chong, 2001), which incorporates
reinforcement and belief learning. It modifies EWA, above,
replacing free parameters φ, δ, and κ with φi(t), δi(t), and
κi(t), functions of the player’s experience so far.
Nick Montfort
nickm@cis.upenn.edu
University of Pennsylvania, Computer and Information Science